Invariance conditions for gauge fields under smooth group actions are interpreted in terms of invariant connections on principal bundles. A classification bundles as automorphisms projecting to an action a base manifold with sufficiently regular orbit structure is given homorphisms and generalization Wang’s theorem classifying derived. Illustrative examples compactified Minkowski space given.

A superposition rule is obtained for the matrix Riccati equation (MRE) Ẇ=A+WB+CW+WDW [where W(t), A(t), B(t), C(t), and D(t) are real n×n matrices], expressing general solution in terms of five known solutions all n≥2. The symplectic MRE (W=WT, A=AT, D=DT, C=BT) treated separately, a derived involving only four solutions. For ‘‘unitary’’ GL(n,R) subcases (with D=A C=BT, or D=−A respectively), rules two derivation these results based upon an interpretation action groups SL(2n,R), SP(2n,R),...

Symmetry reduction is studied for the relativistically invariant scalar partial differential equation H(⧠u,(∇u)2,u)=0 in (n+1)-dimensional Minkowski space M(n,1). The introduction of k symmetry variables ξ1, ... ,ξk as invariants a subgroup G Poincaré group P(n,1), having generic orbits codimension k≤n M(n,1), reduces to PDE variables. All codimension-1 M(n,1) (n arbitrary), reducing an ODE are found, well all codimension-2 and -3 low-dimensional cases n=2,3. type includes many physical...

This work is concerned with the characterization of tensor fields in (compactified) Minkowski space which are invariant under action subgroups conformal group. The general method for determining all smooth a Lie group G on manifold M given, both global and local form. maximal divided into conjugacy classes Poincaré most 1-forms, 2-forms, symmetric (0,2) tensors scalar densities representatives each class (as well as certain other subgroups) then determined. results discussed from viewpoint...

Double Hurwitz numbers enumerating weighted n-sheeted branched coverings of the Riemann sphere or, equivalently, paths in Cayley graph Sn generated by transpositions are determined an associated weight generating function. A uniquely 1-parameter family 2D Toda τ-functions hypergeometric type is shown to consist functions for such numbers. Four classical cases detailed, which weighting uniform: Okounkov’s double ramification simple at all but two specified branch points; case Belyi curves,...

In this work the SU(2) Yang–Mills equations are studied in compactified Minkowski space. The manifold is identified with that of Lie group U(1) ×SU(2) and a classification made all principal bundles over base space terms homotopy classes mappings f:S3→S3. Invariance gauge fields under transformation groups defined bundle case invariance translations shown to imply trivial structure. All solutions field invariant obtained as well (anti-) self-dual translations.